[1]} prove that for any valid weighting function, if the state aggregation function satisfies Eq. REF , then the suboptimality of \(\pi ^{GA}\)  in the ground MDP is bounded by a function linear in \(\varepsilon \) :
\(V_{\pi ^*}(\mathbf {x}) - V_{\pi ^{GA}}(\mathbf {x}) \le \mathcal {O}(\varepsilon ), \textnormal { for all }\mathbf {x} \in \mathbf {X}\) 
